Step |
Hyp |
Ref |
Expression |
1 |
|
xrssre.1 |
|- ( ph -> A C_ RR* ) |
2 |
|
xrssre.2 |
|- ( ph -> -. +oo e. A ) |
3 |
|
xrssre.3 |
|- ( ph -> -. -oo e. A ) |
4 |
|
ssxr |
|- ( A C_ RR* -> ( A C_ RR \/ +oo e. A \/ -oo e. A ) ) |
5 |
1 4
|
syl |
|- ( ph -> ( A C_ RR \/ +oo e. A \/ -oo e. A ) ) |
6 |
|
3orass |
|- ( ( A C_ RR \/ +oo e. A \/ -oo e. A ) <-> ( A C_ RR \/ ( +oo e. A \/ -oo e. A ) ) ) |
7 |
5 6
|
sylib |
|- ( ph -> ( A C_ RR \/ ( +oo e. A \/ -oo e. A ) ) ) |
8 |
7
|
orcomd |
|- ( ph -> ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) ) |
9 |
2 3
|
jca |
|- ( ph -> ( -. +oo e. A /\ -. -oo e. A ) ) |
10 |
|
ioran |
|- ( -. ( +oo e. A \/ -oo e. A ) <-> ( -. +oo e. A /\ -. -oo e. A ) ) |
11 |
9 10
|
sylibr |
|- ( ph -> -. ( +oo e. A \/ -oo e. A ) ) |
12 |
|
df-or |
|- ( ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) <-> ( -. ( +oo e. A \/ -oo e. A ) -> A C_ RR ) ) |
13 |
12
|
biimpi |
|- ( ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) -> ( -. ( +oo e. A \/ -oo e. A ) -> A C_ RR ) ) |
14 |
8 11 13
|
sylc |
|- ( ph -> A C_ RR ) |